Dinamica Clases 7 y 8

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1/40

VECTOR MECHANICS FOR ENGINEERS:

DYNAMICS

Eighth Edition

Ferdinand P. Beer

E. Russell Johnston, Jr.

Leture Notes!

J. "alt #ler

$e%as $eh &ni'ersit(

CHAPTER

2007 The McGraw-Hill Companies, Inc. All rights

13Cintica de Partculas:

Metodos de Energa

and Cantidad de

Movimiento.


2/40

2007 The McGraw-Hill Companies, Inc. All rights reserve.

)etor Mehanis *or Engineers! D(na+isEighth

- /

ContentsIntroduction

Work of a ForcePrinciple of Work & Energy

Applications of the Principle of Work

& Energy

Power and Efficiency

Sample Problem 1!1Sample Problem 1!"

Sample Problem 1!

Sample Problem 1!#

Sample Problem 1!$

Potential Energy

%onseratie Forces

%onseration of Energy

'otion (nder a %onseratie %entral

Force

Sample Problem 1!)

Sample Problem 1!*Sample Problem 1!+

Principle of Impulse and 'omentum

Impulsie 'otion

Sample Problem 1!1,

Sample Problem 1!11

Sample Problem 1!1"

Impact

-irect %entral Impact

.bli/ue %entral Impact

Problems Inoling Energy and 'omentum

Sample Problem 1!1#

Sample Problem 1!1$

Sample Problems 1!1)

Sample Problem 0!1*


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- -

Introduction

Preiously2 problems dealing with the motion of particles were

soled through the fundamental e/uation of motion2%urrent chapter introduces two additional methods of analysis!

!amF=

Method of work and energy3 directly relates force2 mass2

elocity and displacement!

Method of impulse and momentum3 directly relates force2

mass2 elocity2 and time!


4/40



- 0

Work o a !orce

-ifferential ector is theparticle displacement!rd

Work of the forceis

dzFdyFdxF

dsF

rdFdU

zyx ++=

=

=

cos

Work is a scalar /uantity2 i!e!2 it has magnitude and

sign but not direction!

force!length -imensions of work are (nits are( ) ( )( ) 41!$)lb1ftm15141 ==joule

E


5/40



- 1

Work o a !orce

Work of a force during a finite displacement2

( )

( )

++=

==

=

"

1

"

1

"

1

"

1

cos

"1

A

Azyx

s

s

t

s

s

A

A

dzFdyFdxF

dsFdsF

rdFU

Work is represented by the area under thecure ofFtplotted againsts!

E


6/40



- 2

Work o a !orce

Work of a constant force in rectilinear motion2

( ) xFU = cos"1

Work of the force of graity2

( ) yWyyW

dyWU

dyW

dzFdyFdxFdU

y

y

zyx

==

=

=

++=

1"

"1

"

1

Work of the weightis e/ual to product of

weight Wand ertical displacement y.

Work of the weightis positie when y < ,2i!e!2 when the weight moes down!

E


7/40



- 3

Work o a !orce 'agnitude of the force e6erted by a spring is

proportional to deflection2

( )lb7in!or57mconstantspring==

k

kxF

Work of the force exerted y spring2

"""

1"1"

1"1

"

1

kxkxdxkxU

dxkxdxFdU

x

x

==

==

Work of the force exerted y springis positie

whenx"8x

12 i!e!2 when the spring is returning to

its undeformed position!

Work of the force e6erted by the spring is e/ual to

negatie of area under cure ofFplotted against

x2 ( ) xFFU += "1"1

"1

E


8/40 2007 The McGraw-Hill Companies, Inc. All rights reserve.


- 4

Work o a !orce

Work of a graitational force 9assume particleMoccupies fi6ed position !while particle mfollows path

shown:2

1"""1

"

"

1r

Mm"

r

Mm"dr

r

Mm"U

drr

Mm"FdrdU

r

r

==

==

E




- 5

Work o a !orce

Forces which do notdo work #ds; , or cos = 0)3

weight of a body when its center of graity moes

hori




- 6

Particle "inetic Energ#: Princi$le o Work % Energ#

d$m$dsF

ds

d$m$

dt

ds

ds

d$m

dt

d$

mmaF

t

tt

=

==

==

%onsider a particle of mass m acted upon by forceF

Integrating fromA%toA&2

energykineticm$'''U

m$m$d$$mdsF$

$

s

st

===

==

""1

1""1

"1"

1"""

1"

1

"

1

'he work of the force is e(ual to the change in

kinetic energy of the particle!

F

(nits of work and kinetic energy are the same3

4m5m

s

mkg

s

mkg

"

""

"

1 ==

=

== m$'

) t M h i * E i D iE




-

A$$lications o t&e Princi$le o Work and Energ#

Wish to determine elocity of pendulum bob

atA"! %onsider work & kinetic energy!

Force acts normal to path and does no

work!

)

gl$

$g

WWl

'U'

"

"

1,

"

"

"

""11

=

=+

=+

=elocity found without determining

e6pression for acceleration and integrating!

All /uantities are scalars and can be added

directly!

Forces which do no work are eliminated

from the problem!





- /

A$$lications o t&e Princi$le o Work and Energ#

Principle of work and energy cannot be

applied to directly determine the accelerationof the pendulum bob!

%alculating the tension in the cord re/uires

supplementing the method of work and

energy with an application of 5ewton>s

second law! As the bob passes throughA&2

Wl

gl

g

WW)

l

$

g

W

W)

amF nn

"

""

=+=

=

=

gl$ ""=





- -

Po'er and Eicienc#

rate at which work is done!

$F

dt

rdF

dt

dU

)ower

=

==

=

-imensions of power are work7time or force?elocity!

(nits for power are

W*#)s

lbft$$,hp1or

s

m51

s

419watt:W1 =

===

inputpower

outputpower

input work

koutput wor

efficiency

=

=

=





- 0

(am$le Pro)lem *+.*

An automobile weighing #,,, lb is

drien down a $oincline at a speed of

), mi7h when the brakes are applied

causing a constant total breaking forceof 1$,, lb!

-etermine the distance traeled by the

automobile as it comes to a stop!

S.@(I.53

Ealuate the change in kinetic energy!

-etermine the distance re/uired for the

work to e/ual the kinetic energy change!





- 1

(am$le Pro)lem *+.*S.@(I.53

Ealuate the change in kinetic energy!

( ) ( ) lbft#B1,,,BB"!"#,,,

sftBBs),,

h

mi

ft$"B,

h

mi),

"

"1"

1"1

1

1

===

=

=

m$'

$

( ) ,lb11$1lbft#B1,,,""11

==+

x

'U'

ft#1B=x

-etermine the distance re/uired for the work

to e/ual the kinetic energy change!

( ) ( ) ( )

( )x

xxU

lb11$1

$sinlb#,,,lb1$,,"1 =

+=

,, "" == '$

) t M h i * E i D iEi




- 2

(am$le Pro)lem *+.,

wo blocks are Coined by an ine6tensible

cable as shown! If the system is released

from rest2 determine the elocity of block

Aafter it has moed " m! Assume that

the coefficient of friction between blockAand the plane isk; ,!"$ and that the

pulley is weightless and frictionless!

S.@(I.53

Apply the principle of work and

energy separately to blocksAand*!

When the two relations are combined2

the work of the cable forces cancel!

Sole for the elocity!





- 3

(am$le Pro)lem *+.,S.@(I.53

Apply the principle of work and energy separatelyto blocksAand*!

( )( )( )

( ) ( )

( ) ( ) ( ) ( ) ""1

""1

""11

"

kg",,m"5#+,m"

m"m",

3

5#+,51+)""$!,

51+)"smB1!+kg",,

$F

$mFF

'U'

W+F

W

,

AA,

AkAkA

A

=

=+=+

======

( )

( ) ( )

( ) ( ) ( ) ( ) ""1

""1

""11

"

kg,,m"5"+#,m"

m"m",

35"+#,smB1!+kg,,

$F

$mWF

'U'W

c

**c

*

=+

=+

=+==




)etor Mehanis *or Engineers! D(na+isighth

- 4

(am$le Pro)lem *+., When the two relations are combined2 the work of the

cable forces cancel! Sole for the elocity!( ) ( ) ( ) ( ) "

"1 kg",,m"5#+,m" $F, =

( ) ( ) ( ) ( ) ""1 kg,,m"5"+#,m" $Fc =+

( ) ( ) ( ) ( ) ( )

( ) ""1

""1

kg$,,4#+,,

kg,,kg",,m"5#+,m"5"+#,

$

$

=

+=

sm#!#=$

)etor Mehanis *or Engineers D na+isEi



)etor Mehanis *or Engineers! D(na+isghth

- 5

(am$le Pro)lem *+.+

A spring is used to stop a ), kg package

which is sliding on a hori




- /6

(am$le Pro)lem *+.+S.@(I.53

Apply principle of work and energy between initial

position and the point at which spring is fully compressed!

( ) ( ) ,4$!1B*sm$!"kg), ""

"1"

1"1

1 ==== 'm$'

( )

( ) ( )( ) ( ) kk

kfxWU

4**m)#,!,smB1!+kg),"

"1

==

=

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) 4,!11"m,#,!,5",,5"#,,

5",,m1),!,mk5",

5"#,,m1",!,mk5",

"1

ma6min"1

"1

,ma6

,min

=+=

+=

==+====

x))U

xxk)

kx)

e

( ) ( ) ( ) 411"4**"1"1"1 =+= kef UUU

( ) ,411"4**D4$!1B*

3""11

=

=+

k

'U'

",!,

=k

)etor Mehanis *or Engineers! D(na+isEig




- /

(am$le Pro)lem *+.+ Apply the principle of work and energy for the rebound

of the package!

( ) ""1"

"1

" kg),, $m$'' ===

( ) ( ) ( )

4)!$

411"4**"""

+=

+=+= kef UUU

( ) ""1

""

kg),4$!),

3

$

'U'

=+

=+

sm1,!1=$





- //

(am$le Pro)lem *+.-

A ",,, lb car starts from rest at point 1

and moes without friction down the

track shown!

-etermine3

a: the force e6erted by the track on

the car at point "2 and

b: the minimum safe alue of the

radius of curature at point !

S.@(I.53

Apply principle of work and energy to

determine elocity at point "!

Apply 5ewton>s second law to find

normal force by the track at point "!

Apply principle of work and energy todetermine elocity at point !

Apply 5ewton>s second law to find

minimum radius of curature at point

such that a positie normal force is

e6erted by the track!





- /-

(am$le Pro)lem *+.-S.@(I.53

Apply principle of work and energy to determine

elocity at point "!

( )

( )

( ) ( ) ( ) sftB!$,ft"!"ft#,"ft#,""

1ft#,,3

ft#,

"

1,

"""

"

""""11

"1

""

"""

1"1

===

=+=+

+=

===

$sg$

$g

WW'U'

WU

$g

Wm$''

Apply 5ewton>s second law to find normal force by

the track at point "!

3nn amF =+ ( )

W+

g

g

W$

g

Wam+W n

$

ft",

ft#,"

"

""

=

===+

lb1,,,,=+





- /0

(am$le Pro)lem *+.- Apply principle of work and energy to determine

elocity at point !

( )

( ) ( ) ( ) sft1!#,sft"!"ft"$"ft"$"

"

1ft"$,

"

"11

===

=+=+

$g$

$g

WW'U'

Apply 5ewton>s second law to find minimum radius ofcurature at point such that a positie normal force is

e6erted by the track!

3nn amF =+

( )

" ft"$"

g

g

W$

g

W

amWn

===

ft$,=





- /1

(am$le Pro)lem *+.

he dumbwaiterand its load hae a

combined weight of ),, lb2 while the

counterweight , weighs B,, lb!

-etermine the power deliered by theelectric motorMwhen the dumbwaiter

#a-is moing up at a constant speed of

B ft7s and #-has an instantaneous

elocity of B ft7s and an acceleration of

"!$ ft7s"2 both directed upwards!

S.@(I.53

Force e6erted by the motorcable has same direction as

the dumbwaiter elocity!

Power deliered by motor is

e/ual to F$/ $0 B ft7s!

In the first case2 bodies are in uniform

motion! -etermine force e6erted by

motor cable from conditions for static

e/uilibrium!

In the second case2 both bodies are

accelerating! Apply 5ewton>s

second law to each body to

determine the re/uired motor cable

force!



26/40



- /2

(am$le Pro)lem *+. In the first case2 bodies are in uniform motion!

-etermine force e6erted by motor cable from

conditions for static e/uilibrium!

( ) ( )

slbft1),,

sftBlb",,

=== F$)ower

( ) hp+1!"slbft$$,

hp1slbft1),, =

=)ower

FreeDbody %3

3,=+ yF lb#,,,lbB,," == ''

FreeDbody -3

3,=+ yFlb",,lb#,,lb),,lb),,

,lb),,

====+

'F

'F



27/40



- /3

(am$le Pro)lem *+. In the second case2 both bodies are accelerating! Apply

5ewton>s second law to each body to determine the re/uired

motor cable force!

=== ""1" sft"$!1sft$!" , aaa

FreeDbody %3

3,,y amF =+ ( ) lb$!B#"$!1"!"

B,,"B,, == ''

FreeDbody -3

3y amF =+ ( )

lb1!")")!#)),,$!B#

$!""!"

),,),,

==+

=+

FF

'F

( ) ( ) slbft",+*sftBlb1!")" === F$)ower

( ) hpB1!

slbft$$,

hp1slbft",+* =

=)ower



28/40



- /4

Potential Energ#

"1"1 yWyWU =

Work of the force of graity 2W

Work is independent of path followed depends

only on the initial and final alues of Wy.

=

=Wy1g

potential energyof the body with respect

toforce of gra$ity!

"1"1 gg 11U =

(nits of work and potential energy are the same3

4m5 === Wy1g

%hoice of datum from which the eleationyismeasured is arbitrary!



29/40



- /5

Potential Energ#

Preious e6pression for potential energy of a body

with respect to graity is only alid when theweight of the body can be assumed constant!

For a space ehicle2 the ariation of the force of

graity with distance from the center of the earth

should be considered!

Work of a graitational force2

1""1

r

"Mm

r

"MmU =

Potential energy 1gwhen the ariation in the

force of graity can not be neglected2

r

W2

r

"Mm1g

"

==



30/40


)etor Mehanis *or Engineers! D(na+ishth

- -6

Potential Energ#

Work of the force e6erted by a spring depends

only on the initial and final deflections of thespring2

"

""

1"

1"

1"1 kxkxU =

he potential energy of the body with respect

to the elastic force2

( ) ( )"1"1

""1

ee

e

11U

kx1

=

=

5ote that the preceding e6pression for 1eis

alid only if the deflection of the spring is

measured from its undeformed position!



31/40



- -

Conservative !orces %oncept of potential energy can be applied if the

work of the force is independent of the path

followed by its point of application!

( ) ( )"""111"1 2222 zyx1zyx1U =Such forces are described as conser$ati$e forces!

For any conseratie force applied on a closed path2

,= rdF Elementary work corresponding to displacement

between two neighboring points2

( ) ( )

( )zyxd1

dzzdyydxx1zyx1dU

22

2222

=

+++=

1z

1

y

1

x

1F

dzz

1dy

y

1dx

x

1dzFdyFdxF zyx

grad=

+

+

=

+

+

=++

)etor Mehanis *or Engineers! D(na+isEigh


32/40



- -/

Conservation o Energ# Work of a conseratie force2

"1"1

11U =

%oncept of work and energy2

1""1 ''U =

Follows that

constant

""11

=+=+=+

1'3

1'1'

When a particle moes under the action of

conseratie forces2 the total mechanical

energy is constant!

W1'

W1'

=+==

11

11 ,

( )

W1'

1WggWm$'

=+

====

""

""""1" ,"

"1 Friction forces are not conseratie! otal

mechanical energy of a system inoling

friction decreases!

'echanical energy is dissipated by friction

into thermal energy! otal energy is constant!



33/40



- --

Motion /nder a Conservative Central !orce When a particle moes under a conseratie central

force2 both the principle of conseration of angular

momentum

and the principle of conseration of energy

may be applied!

sinsin ,,, rm$m$r =

r

"Mmm$

r

"Mmm$

1'1'

=

+=+

""1

,

","

1

,,

ien r2 the e/uations may be soled for $and .

At minimum and ma6imum r2 = +,o! ien the

launch conditions2 the e/uations may be soled for

rmin/ rmax/ $min2 and $max!



34/40



- -0

(am$le Pro)lem *+.0

A ", lb collar slides without friction

along a ertical rod as shown! he

spring attached to the collar has an

undeflected length of # in! and aconstant of lb7in!

If the collar is released from rest at

position 12 determine its elocity after

it has moed ) in! to position "!

S.@(I.53

Apply the principle of conseration of

energy between positions 1 and "!

he elastic and graitational potential

energies at 1 and " are ealuated from

the gien information! he initialkinetic energy is


35/40



- -1

(am$le Pro)lem *+.0S.@(I.53

Apply the principle of conseration of energy between

positions 1 and "!

Position 13 ( ) ( )

,

lbft",lbin!"#

lbin!"#in!#in!Bin!lb

1

1

"

"1"

1"1

=

=+=+=

===

'

111

kx1

ge

e

Position "3 ( ) ( )

( ) ( )

""

""

""

"

1"

"

""1"

""1

11!,

"!"

",

"

1

lbft$!$lbin!))1",$#

lbin!1",in!)lb",

lbin!$#in!#in!,1in!lb

$$m$'

111

Wy1

kx1

ge

g

e

===

===+=

===

===

%onseration of Energy3

lbft$!$,!11lbft", ""

""11

=+

+=+

$

1'1'

= sft+1!#"$



36/40



- -2

(am$le Pro)lem *+.1

he ,!$ lb pellet is pushed against the

spring and released from rest atA!

5eglecting friction2 determine thesmallest deflection of the spring for

which the pellet will trael around the

loop and remain in contact with the

loop at all times!

S.@(I.53

Since the pellet must remain in contactwith the loop2 the force e6erted on the

pellet must be greater than or e/ual to


37/40



- -3

(am$le Pro)lem *+.1S.@(I.53

Setting the force e6erted by the loop to


38/40



- -4

(am$le Pro)lem *+.2

A satellite is launched in a direction

parallel to the surface of the earth

with a elocity of )+,, km7h from

an altitude of $,, km!

-etermine #a-the ma6imum altitudereached by the satellite2 and #-the

ma6imum allowable error in the

direction of launching if the satellite

is to come no closer than ",, km to

the surface of the earth

S.@(I.53

For motion under a conseratie centralforce2 the principles of conseration of

energy and conseration of angular

momentum may be applied simultaneously!

Apply the principles to the points of

minimum and ma6imum altitude todetermine the ma6imum altitude!

Apply the principles to the orbit insertion

point and the point of minimum altitude to

determine ma6imum allowable orbit

insertion angle error!



39/40



- -5

(am$le Pro)lem *+.2 Apply the principles of conseration of energy and

conseration of angular momentum to the points of minimum

and ma6imum altitude to determine the ma6imum altitude!

%onseration of energy3

1

"1"

1

,

","

1

r

"Mmm$

r

"Mmm$1'1' AAAA =+=+

%onseration of angular momentum3

1,,111,,r

r$$m$rm$r ==

%ombining2

",,1

,

1

,

,"

1

","

,"1 "111

$r

"M

r

r

r

r

r

"M

r

r$ =+

=

( )( ) "1"")""

),

,

sm1,+Bm1,*!)smB1!+

sm1,"$!1,hkm)+,,

km)B*,km$,,km)*,

===

===+=

g2"M

$

r

km),#,,m1,#!), )

1

==r



40/40

)etor Mehanis *or Engineers! D(na+isth

(am$le Pro)lem *+.2 Apply the principles to the orbit insertion point and the point

of minimum altitude to determine ma6imum allowable orbit

insertion angle error!

%onseration of energy3

min

"ma6"

1

,

","

1,,

r

"Mmm$

r

"Mmm$1'1' AA =+=+

%onseration of angular momentum3

min

,,,ma6ma6min,,, sinsin

rr$$m$rm$r ==

%ombining and soling for sin02

=

=

$!11+,

+B,1!,sin

,

,

= $!11errorallowable

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